Prime-power congruences for level-one eigenforms

Abstract: We study prime-power congruences for the prime-to-(p) Hecke eigensystems of normalized cuspidal level-one eigenforms as the weight varies, and we address the problem of classifying such systems for small prime powers. We construct Hecke-equivariant transfer maps between level-one modular-symbol modules via multiplication by suitable lifts of Dickson invariants; these maps reduce the problem to finite computation and propagate Hecke operator relations through all weights. For (p=2,3,5,7), we obtain all-weight classifications modulo (2^8), (3^4), (5^3), and (7^2), respectively.